Positional parameter

The first crystal structure to be detennined that had an adjustable position parameter was that of pyrite, FeS2 In this structure the iron atoms are at the comers and the face centres, but the sulphur atoms are further away than in zincblende along a different tln-eefold synnnetry axis for each of the four iron atoms, which makes the unit cell primitive. 

Let us construct the standard penalty operator (5 v) = I v — Pv) and define the penalty problem depending on a small positive parameter s,... 

Proof. We introduce the penalty operator p w) = — w — ) and consider the auxiliary boundary value problem with the positive parameter e > 0,... 

Proof. Let s, 5 be positive parameters, and p be the penalty operator introduced in the previous section. We consider the auxiliary problem... 

Proof. Using elliptic regularization and the penalty approach, we construct an auxiliary problem which approximates (5.6)-(5.9). Its solution will depend on two positive parameters a and d which are related to the elliptic regularization and to the penalty approach, respectively. We will obtain a solution a, u by passing to the limit as a, (5 —> 0. So, consider the following boundary value problem in fl... 

Proof. We consider a parabolic regularization of the problem approximating (5.68)-(5.72). The auxiliary boundary value problem will contain two positive parameters a, 5. The first parameter is responsible for the parabolic regularization and the second one characterizes the penalty approach. Our aim is first to prove an existence of solutions for the fixed parameters a, 5 and second to justify a passage to limits as a, d —> 0. A priori estimates uniform with respect to a, 5 are needed to analyse the passage to the limits, and we shall obtain all necessary estimates while the theorem of existence is proved. 

Proof. The idea of the proof is to use an elliptic regularization for the penalty equations approximating (5.139)-(5.143). Solutions of the auxiliary problem will depend on two positive parameters s, 5. The first parameter is responsible for the elliptic regularization and the second one characterizes... 

Assume that p n,m) = n,m) — 7r(n,m). Consider the auxiliary boundary problem containing three positive parameters s, 5, A without stating the dependence of the solution on these parameters ... 

We introduce three positive parameters s, 5, A and consider the regularized problem... 

Proof. Let the operators ( i, 2), P be the same as in the preceding section. In the domain Q we consider the auxiliary boundary value problem with three positive parameters e, 5, A,... 

Step 11. At this point a computer program refines the atomic parameters of the atoms that were assigned labels. The atomic parameters consist of the three position parameters x,j, and for each atom. Also one or six atomic displacement parameters that describe how the atom is "smeared" (due to thermal motion or disorder) are refined for each atom. The atomic parameters are varied so that the calculated reflection intensities are made to be as nearly equal as possible to the observed intensities. During this process, estimated phase angles are obtained for all of the reflections whose intensities were measured. A new three-dimensional electron density map is calculated using these calculated phase angles and the observed intensities. There is less false detail in this map than in the first map. 

Zachariasen s arrangement of the metal atoms approximates the first of our two (that with the positive parameter value), and would be identical with it if his parameters were taken to be 0.030 and 0.530 rather than 0.021 and 0.5 2. 

In Table 2 there are given data pertinent to the lattice-parameter determination, as well as intensity data used in the determination of atomic positional parameters. 

The reliability factor B was 0276 after the first refinement and 0-211 after the fourth refinement. The parameters from the third and fourth refinements differed very little from one another. The final values are given in Table 1. As large systematic errors were introduced in the refinement process by the unavoidable use of very poor atomic form factors, the probable errors in the parameters as obtained in the refinement were considered to be of questionable significance. For this reason they are not given in the table. The average error was, however, estimated to be 0-001 for the positional parameters and 5% for the compositional parameters. The scattering power of the two atoms of type A was given by the least-squares refinement as only 0-8 times that of aluminum (the fraction... 

This structure, based on the space group OjtPm in, involves about fifty atomic positional parameters, for which approximate values can be estimated from the Mg32(Al,Zn)49 parameters. Refinement of the structure would be laborious. 

Positional parameters of the non-hydrogen atoms obtained from refinements I and II are in good agreement with those of SC (1980) or Dam, Harkema and Feil (hereafter DHF) [16] from X-ray data as well as those from neutron data [13, 17]. 

In the final stages of the refinement the positional parameters of the H atoms were kept fixed, and these atoms too were described with multipoles, up to the dipole level. For both poles of the H pseudoatoms the radial functions were again single exponentials, with n = 0, 1 for monopole and dipole respectively, and the a value was 2.48 bohr1. 

FeSa—m type structure confirm that Hyp. 6 is approximately satisfied, the ratio (a2 +c2)/62 varying between 0.995 (CrSba) and 1.035 (FeSba) in class A, and between 1.023 (CuSea) and 1.093 (CoTea) in class B. The situation concerning Hyp. 7 is similar to that for Hyp. 5 but, since FeSa is the only compound for which accurate positional parameters are available for both modifications xv —0.3840(5) and y=0.37820(5), cf. (5, 6)], further experimental tests of the degree of validity of this postulate are called for. Note that the less accurate parameters for NaC>2 satisfy (8) the relation exactly (xp =y=0.43). 

Refinement takes place by adjusting the model to find closer agreement between the calculated and observed structure factors. For proteins the refinements can yield R-factors in the range of 10-20%. An example taken from reference 10 is instructive. In a refinement of a papain crystal at 1.65-A resolution, 25,000 independent X-ray reflections were measured. Parameters to be refined were the positional parameters (x, y, and z) and one isotropic temperature factor parameter... 

Modifications separated by a second-order transition can never be coexistent. One typical second-order transition, the displacive structural transition, is characterized by the distortion of bonds rather than their breaking, and the structural changes that occur are usually small. Typically, there is continuous variation in the positional parameters and the unit cell dimensions as a function of temperature. The structural changes in the system occur gradually as the system moves away from the transition point. As well as a structural similarity, a symmetry relationship... 

An example of inverse calculation from dissolved Ni concentrations in the Eastern Pacific measured by Bruland (1980) is discussed in Chapter 5. Particularly important in inverting the data is to make sure that e must be larger than unity, since the rate constant is a positive parameter, o... 

Considering Au in 0, 0, 0 as the reference atom, the next neighbours Au atoms are the six Au shown in Fig. 3.29(a), corresponding to the same Wyckoff position and having, in comparison with the reference atom, the coordinates 0, 0, 1 0, 0, 1 0, 1, 0 0, 1, 0 1, 0, 0 1, 0, 0, all at a distance d = a = 374.8 pm, that is at a reduced distance dr = d/dmin = 1.414. Notice that in the analysis of the structure it may be necessary to consider not only the positions of the atoms in the reference cell but also those in the adjacent cells. Notice also that, in a simple cubic structure without free positional parameters such as the AuCu3 type, the reduced distances are independent of the values of the lattice parameters and are the same for all the isostructural compounds. 

In order to have around each atom in this hexagonal structure four exactly equidistant neighbouring atoms, the axial ratio should have the ideal value (8/3 that is 1.633. The experimental values range from 1.59 to 1.66. This practical constancy of the axial ratio, in contrast with what is observed for other families of isostructural compounds such as those of the NiAs type, may be attributed to a sort of rigidity of the tetrahedral (sp3) chemical bonds. As for the atomic positional parameters, the ideal value of one of the parameters (being the other one fixed at zero by conventionally shifting the origin of the cell) is z = 3/8 = 0.3750. The C diamond, sphalerite- and wurtzite-type structures are well-known examples of the normal tetrahedral structures (see 3.9.2.2). 

The structure of KHg2 may be recognized as isotypic with that of CeCu2, even if this is not immediately apparent from the data (axes orientation, positional parameters) reported in the original structure determinations and, consequently, in various compilations. See a few comments in the following. 

Table 2. Atomic positional parameters estimated from the reconstructed exit wave, those obtained after the refinement with all Cu atoms and subsequently refined occupancies...

Table 3. Final atomic position parameters for Ce5Cuj9Pj2 determined by MSLS and for La5Cuj9Pj2 determined X-ray single crystal diffraction data.

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